Derived-series growth of the Nottingham group (${\log }_p[\mathcal{N}:{\mathcal{N}}^{(n)}]=O({\lambda }^n),\lambda <3$)

For odd $p$, group commutator collection in the depth filtration gives $[{\mathcal{N}}_m,{\mathcal{N}}_m]={\mathcal{N}}_{2m+1}$. Every degree $k\ge 2m+1$ occurs as a leading group commutator degree by choosing $r,s\ge m$, $r+s=k$, and $r\not\equiv s(\mathrm{mod}p)$; changing $r$ by one changes $r-s$ by two. Closed successive coefficient correction then fills the tail. Therefore ${\mathcal{N}}^{(n)}={\mathcal{N}}_{2^{n+1}-1}$ and ${\log }_p[\mathcal{N}:{\mathcal{N}}^{(n)}]=2^{n+1}-2$. In characteristic two the sharper collection theorem gives ${\log }_2[\mathcal{N}:{\mathcal{N}}^{(n)}]\le C{\kappa }^n$ for a constant $C$, with $\kappa =(1+\sqrt{17})/2<3$. In both cases the exponential base is strictly smaller than three.